@HJ: There are misprints in the definition of $G$. Could you please correct them? Is it supposed to be an LOG presentation?
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Mark SapirDec 8 '10 at 21:36

HJ - as stated, this is just a free group. (Killing a conjugate of $x_i$ is the same as killing $x_i$.) I presume you're aware of Magnus's Theorem that free groups are residually nilpotent (in particular, residually solvable).
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HJRWDec 8 '10 at 21:42

@Henry, it is a clear misprint. He probably meant an LOG presentation.
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Mark SapirDec 8 '10 at 21:53

@Mark, Yes, I corrected the presentation. There were missing $x_[i+1}$.
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hjjangDec 9 '10 at 20:29

It is not clear whether the set of finite residually solvable group presentations is even recursively enumerable. Unlike for the word or triviality problem where there exists an algorithm which says "yes" iff the answer is "yes", I do not think there exists such an algorithm in this case. But I do not think anybody proved that the algorithm does not exist. Same for residually finite groups.